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Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

  • Laurette S. TuckermanEmail author
  • Jacob Langham
  • Ashley Willis
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 50)

Abstract

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton’s method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier–Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10–50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.

Notes

Acknowledgements

We thank Dwight Barkley and John Gibson for their contributions. We acknowledge the support of TRANSFLOW, provided by the Agence Nationale de la Recherche (ANR).

References

  1. 1.
    Tuckerman, L.S.: Steady-state solving via Stokes preconditioning; recursion relations for elliptic operators. In: Voigt, R.G., Dwoyer, D.L., Hussaini, M.Y. (eds.) 11th International Conference on Numerical Methods in Fluid Dynamics, pp. 573–577. Springer, Berlin (1989)CrossRefGoogle Scholar
  2. 2.
    Mamun, C.K., Tuckerman, L.S.: Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Xin, S., Le Quéré, P., Daube, O.: Natural convection in a differentially heated horizontal cylinder: effects of Prandtl number on flow structure and instability. Phys. Fluids 9, 1014–1033 (1997)CrossRefGoogle Scholar
  4. 4.
    Xin, S., Le Quéré, P.: An extended Chebyshev pseudo-spectral benchmark for the 8:1 differentially heated cavity. Int. J. Numer. Methods Fluids 40, 981–998 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Xin, S., Le Quéré, P.: Stability of two-dimensional (2D) natural convection flows in air-filled differentially heated cavities: 2D/3D disturbances. Fluid Dyn. Res. 44, 031419 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chénier, E., Delcarte, C., Labrosse, G.: Stability of the axisymmetric buoyant-capillary flows in a laterally heated liquid bridge. Phys. Fluids 11, 527–541 (1999)zbMATHCrossRefGoogle Scholar
  7. 7.
    Mercader, I., Batiste, O., Ramírez-Piscina, L., Ruiz, X., Rüdiger, S., Casademunt, J.: Bifurcations and chaos in single-roll natural convection with low Prandtl number. Phys. Fluids 17, 104108 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Henry, D., BenHadid, H.: Multiple flow transitions in a box heated from the side in low-Prandtl-number fluids. Phys. Rev. E 76, 016314 (2007)CrossRefGoogle Scholar
  9. 9.
    Dridi, W., Henry, D., Ben Hadid, H.: Influence of acoustic streaming on the stability of a laterally heated three-dimensional cavity. Phys. Rev. E 77, 046311 (2008)CrossRefGoogle Scholar
  10. 10.
    Torres, J.F., Henry, D., Komiya, A., Maruyama, S., Ben Hadid, H.: Three-dimensional continuation study of convection in a tilted rectangular enclosure. Phys. Rev. E 88, 043015 (2013)CrossRefGoogle Scholar
  11. 11.
    Torres, J.F., Henry, D., Komiya, A., Maruyama, S.: Bifurcation analysis of steady natural convection in a tilted cubical cavity with adiabatic sidewalls. J. Fluid Mech. 756, 650–688 (2014)CrossRefGoogle Scholar
  12. 12.
    Torres, J.F., Henry, D., Komiya, A., Maruyama, S.: Transition from multiplicity to singularity of steady natural convection in a tilted cubical enclosure. Phys. Rev. E 92, 023031 (2015)CrossRefGoogle Scholar
  13. 13.
    Touihri, R., BenHadid, H., Henry, D.: On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case. Phys. Fluids 11, 2078–2088 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Touihri, R., El Gallaf, A., Henry, D., BenHadid, H.: Instabilities in a cylindrical cavity heated from below with a free surface. I. Effect of Biot and Marangoni numbers. Phys. Rev. E 84, 056302 (2011)CrossRefGoogle Scholar
  15. 15.
    Assemat, P., Bergeon, A., Knobloch, E.: Nonlinear Marangoni convection in circular and elliptical cylinders. Phys. Fluids 19, 104101 (2007)zbMATHCrossRefGoogle Scholar
  16. 16.
    Marques, F., Mercader, I., Batiste, O., Lopez, J.M.: Centrifugal effects in rotating convection. J. Fluid Mech. 580, 303–318 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Borońska, K., Tuckerman, L.S.: Extreme multiplicity in cylindrical Rayleigh–Bénard convection. II. Bifurcation diagram and symmetry classification. Phys. Rev. E 81, 036321 (2010)CrossRefGoogle Scholar
  18. 18.
    Mercader, I., Sánchez, O., Batiste, O.: Secondary flows in a laterally heated horizontal cylinder. Phys. Fluids 26, 014104 (2014)CrossRefGoogle Scholar
  19. 19.
    Sánchez, O., Mercader, I., Batiste, O., Alonso, A.: Natural convection in a horizontal cylinder with axial rotation. Phys. Rev. E 93, 063113 (2016)CrossRefGoogle Scholar
  20. 20.
    Bergemann, K., Feudel, F., Tuckerman, L.S.: Geoflow: on symmetry-breaking bifurcations of heated spherical shell convection. J. Phys. Conf. Ser. 137, 012027 (2008). (IOP Publishing)CrossRefGoogle Scholar
  21. 21.
    Feudel, F., Bergemann, K., Tuckerman, L.S., Egbers, C., Futterer, B., Gellert, M., Hollerbach, R.: Convection patterns in a spherical fluid shell. Phys. Rev. E 83, 046304 (2011)CrossRefGoogle Scholar
  22. 22.
    Feudel, F., Seehafer, N., Tuckerman, L.S., Gellert, M.: Multistability in rotating spherical shell convection. Phys. Rev. E 87, 023021 (2013)CrossRefGoogle Scholar
  23. 23.
    Feudel, F., Tuckerman, L.S., Gellert, M., Seehafer, N.: Bifurcations of rotating waves in rotating spherical shell convection. Phys. Rev. E 92, 053015 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Feudel, F., Tuckerman, L.S., Zaks, M., Hollerbach, R.: Hysteresis of dynamos in rotating spherical shell convection. Phys. Rev. F 2, 053902 (2017)Google Scholar
  25. 25.
    Daube, O., Le Quéré, P.: Numerical investigation of the first bifurcation for the flow in a rotor-stator cavity of radial aspect ratio 10. Comput. Fluids 31, 481–494 (2002)zbMATHCrossRefGoogle Scholar
  26. 26.
    Nore, C., Tuckerman, L.S., Daube, O., Xin, S.: The 1 [ratio] 2 mode interaction in exactly counter-rotating von kármán swirling flow. J. Fluid Mech. 477, 51–88 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nore, C., Tartar, M., Daube, O., Tuckerman, L.S.: Survey of instability thresholds of flow between exactly counter-rotating disks. J. Fluid Mech. 511, 45–65 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Huepe, C., Metens, S., Dewel, G., Borckmans, P., Brachet, M.-E.: Decay rates in attractive Bose–Einstein condensates. Phys. Rev. Lett. 82, 1616 (1999)CrossRefGoogle Scholar
  29. 29.
    Abid, M., Huepe, C., Metens, S., Nore, C., Pham, C.T., Tuckerman, L.S., Brachet, M.E.: Gross–Pitaevskii dynamics of Bose–Einstein condensates and superfluid turbulence. Fluid Dyn. Res. 33, 509–544 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Xin, S., Le Quéré, P., Tuckerman, L.S.: Bifurcation analysis of double-diffusive convection with opposing horizontal thermal and solutal gradients. Phys. Fluids 10, 850–858 (1998)CrossRefGoogle Scholar
  31. 31.
    Bergeon, A., Henry, D., Benhadid, H., Tuckerman, L.S.: Marangoni convection in binary mixtures with soret effect. J. Fluid Mech. 375, 143–177 (1998)zbMATHCrossRefGoogle Scholar
  32. 32.
    Bergeon, A., Ghorayeb, K., Mojtabi, A.: Double diffusive instability in an inclined cavity. Phys. Fluids 11, 549–559 (1999)zbMATHCrossRefGoogle Scholar
  33. 33.
    Bardan, G., Bergeon, A., Knobloch, E., Mojtabi, A.: Nonlinear doubly diffusive convection in vertical enclosures. Phys. D 138, 91–113 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Bergeon, A., Knobloch, E.: Natural doubly diffusive convection in three-dimensional enclosures. Phys. Fluids 14, 3233–3250 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Bergeon, A., Mollaret, R., Henry, D.: Soret effect and slow mass diffusion as a catalyst for overstability in Marangoni–Bénard flows. Heat Mass Transf. 40, 105–114 (2003)CrossRefGoogle Scholar
  36. 36.
    Meca, E., Mercader, I., Batiste, O., Ramírez-Piscina, L.: Blue sky catastrophe in double-diffusive convection. Phys. Rev. Lett. 92, 234501 (2004)zbMATHCrossRefGoogle Scholar
  37. 37.
    Meca, E., Mercader, I., Batiste, O., Ramírez-Piscina, L.: Complex dynamics in double-diffusive convection. Theor. Comput. Fluid Dyn. 18, 231–238 (2004)zbMATHCrossRefGoogle Scholar
  38. 38.
    Mercader, I., Alonso, A., Batiste, O.: Numerical analysis of the Eckhaus instability in travelling-wave convection in binary mixtures. Eur. Phys. J. E 15, 311–318 (2004)CrossRefGoogle Scholar
  39. 39.
    Batiste, O., Alonso, A., Mercader, I.: Hydrodynamic stability of binary mixtures in Bénard and thermogravitational cells. J. Non-Equilib. Thermodyn. 29, 359–375 (2004)zbMATHCrossRefGoogle Scholar
  40. 40.
    Alonso, A., Batiste, O., Mercader, I.: Numerical analysis of binary fluid convection in extended systems. J. Phys. Conf. Ser 14, 180 (2005). (IOP Publishing)CrossRefGoogle Scholar
  41. 41.
    Alonso, A., Batiste, O., Meseguer, A., Mercader, I.: Complex dynamical states in binary mixture convection with weak negative soret coupling. Phys. Rev. E 75, 026310 (2007)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Burke, J., Knobloch, E.: Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Champneys, A.R.: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys. D 112, 158–186 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Coullet, P., Riera, C., Tresser, C.: Stable static localized structures in one dimension. Phys. Rev. Lett. 84, 3069 (2000)CrossRefGoogle Scholar
  45. 45.
    Fauve, S., Thual, O.: Solitary waves generated by subcritical instabilities in dissipative systems. Phys. Rev. Lett. 64, 282 (1990)CrossRefGoogle Scholar
  46. 46.
    Pomeau, Y.: Front motion, metastability and subcritical bifurcations in hydrodynamics. Phys. D 23, 3–11 (1986)CrossRefGoogle Scholar
  47. 47.
    Hilali, M.F., Métens, S., Borckmans, P., Dewel, G.: Pattern selection in the generalized Swift–Hohenberg model. Phys. Rev. E 51, 2046 (1995)CrossRefGoogle Scholar
  48. 48.
    Batiste, O., Knobloch, E., Alonso, A., Mercader, I.: Spatially localized binary-fluid convection. J. Fluid Mech. 560, 149–158 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Alonso, A., Batiste, O., Mercader, I.: Numerical simulations of binary fluid convection in large aspect ratio annular containers. Eur. Phys. J. Spec. Top. 146, 261–277 (2007)CrossRefGoogle Scholar
  50. 50.
    Bergeon, A., Knobloch, E.: Spatially localized states in natural doubly diffusive convection. Phys. Fluids 20, 034102 (2008)zbMATHCrossRefGoogle Scholar
  51. 51.
    Bergeon, A., Knobloch, E.: Periodic and localized states in natural doubly diffusive convection. Phys. D 237, 1139–1150 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Assemat, P., Bergeon, A., Knobloch, E.: Spatially localized states in Marangoni convection in binary mixtures. Fluid Dyn. Res. 40, 852–876 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    LoJacono, D., Bergeon, A., Knobloch, E.: Spatially localized binary fluid convection in a porous medium. Phys. Fluids 22, 909 (2010)Google Scholar
  54. 54.
    Beaume, C., Bergeon, A., Knobloch, E.: Homoclinic snaking of localized states in doubly diffusive convection. Phys. Fluids 23, 094102 (2011)CrossRefGoogle Scholar
  55. 55.
    Mercader, I., Batiste, O., Alonso, A., Knobloch, E.: Convectons, anticonvectons and multiconvectons in binary fluid convection. J. Fluid Mech. 667, 586–606 (2011)zbMATHCrossRefGoogle Scholar
  56. 56.
    Beaume, C., Bergeon, A., Kao, H.-C., Knobloch, E.: Convectons in a rotating fluid layer. J. Fluid Mech. 717, 417–448 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Beaume, C., Knobloch, E., Bergeon, A.: Nonsnaking doubly diffusive convectons and the twist instability. Phys. Fluids 25, 114102 (2013)CrossRefGoogle Scholar
  58. 58.
    Mercader, I., Batiste, O., Alonso, A., Knobloch, E.: Travelling convectons in binary fluid convection. J. Fluid Mech. 722, 240–266 (2013)zbMATHCrossRefGoogle Scholar
  59. 59.
    LoJacono, D., Bergeon, A., Knobloch, E.: Spatially localized radiating diffusion flames. Combust. Flame 176, 117–124 (2017)CrossRefGoogle Scholar
  60. 60.
    LoJacono, D., Bergeon, A., Knobloch, E.: Localized traveling pulses in natural doubly diffusive convection. Phys. Rev. F 2, 093501 (2017)Google Scholar
  61. 61.
    LoJacono, D., Bergeon, A., Knobloch, E.: Complex convective structures in three-dimensional binary fluid convection in a porous medium. Fluid Dyn. Res. 49, 061402 (2017)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Nagata, M.: Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527 (1990)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Waleffe, F.: Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140 (1998)CrossRefGoogle Scholar
  64. 64.
    Waleffe, F.: Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102 (2001)zbMATHCrossRefGoogle Scholar
  65. 65.
    Waleffe, F.: Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Clever, R.M., Busse, F.H.: Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137–153 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Nagata, M.: Three-dimensional traveling-wave solutions in plane Couette flow. Phys. Rev. E 55, 2023 (1997)CrossRefGoogle Scholar
  68. 68.
    Faisst, H., Eckhardt, B.: Transition from the Couette–Taylor system to the plane Couette system. Phys. Rev. E 61, 7227 (2000)CrossRefGoogle Scholar
  69. 69.
    Schmiegel, A.: Transition to turbulence in linearly stable shear flows. Ph.D. thesis, Philipps-Universität Marburg (1999). http://archiv.ub.uni-marburg.de/diss/z2000/0062/
  70. 70.
    Faisst, H., Eckhardt, B.: Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502 (2003)CrossRefGoogle Scholar
  71. 71.
    Wedin, H., Kerswell, R.R.: Exact coherent structures in pipe flow. J. Fluid Mech. 508, 333–371 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Cvitanović, P.: Periodic orbits as the skeleton of classical and quantum chaos. Phys. D 51, 138–151 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Cvitanovic, P., Eckhardt, B.: Periodic orbit expansions for classical smooth flows. J. Phys. A Math. General 24, L237 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Kawahara, G., Kida, S.: Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Kawahara, G., Uhlmann, M., Van Veen, L.: The significance of simple invariant solutions in turbulent flows. Ann. Rev. Fluid Mech. 44, 203–225 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Dennis Jr., J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations, vol. 16. SIAM, Philadelphia (1996)zbMATHCrossRefGoogle Scholar
  77. 77.
    Knoll, D.A., Keyes, D.E.: Jacobian-free Newton–Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Sánchez, J., Net, M., Garcıa-Archilla, B., Simó, C.: Newton–Krylov continuation of periodic orbits for Navier–Stokes flows. J. Comput. Phys. 201, 13–33 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Viswanath, D.: Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Van Veen, L., Kawahara, G., Atsushi, M.: On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33, 25–44 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Pringle, C.C.T., Kerswell, R.: Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99, 074502 (2007)CrossRefGoogle Scholar
  82. 82.
    Kerswell, R.R., Tutty, O.R.: Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69–102 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Duguet, Y., Willis, A.P., Kerswell, R.R.: Transition in pipe flow. J. Fluid Mech. 613, 255–274 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Duguet, Y., Pringle, C.C.T., Kerswell, R.R.: Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102 (2008)zbMATHCrossRefGoogle Scholar
  85. 85.
    Gibson, J.F., Halcrow, J., Cvitanović, P.: Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Gibson, J.F., Halcrow, J., Cvitanović, P.: Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266 (2009)zbMATHCrossRefGoogle Scholar
  87. 87.
    Pringle, C.C.T., Duguet, Y., Kerswell, R.R.: Highly symmetric travelling waves in pipe flow. Philos. Trans. R. Soc. Lond. A 367, 457–472 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    Gibson, J.F.: Channelflow: a spectral Navier–Stokes simulator in C++. Technical report, University of New Hampshire (2014). www.Channelflow.org
  89. 89.
    Willis, A.P.: The openpipeflow Navier–Stokes solver. SoftwareX 6, 124–127 (2017)CrossRefGoogle Scholar
  90. 90.
    Schneider, T.M., Gibson, J.F., Burke, J.: Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501 (2010)CrossRefGoogle Scholar
  91. 91.
    Avila, M., Mellibovsky, F., Roland, N., Hof, B.: Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502 (2013)CrossRefGoogle Scholar
  92. 92.
    Gibson, J.F., Brand, E.: Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61 (2014)MathSciNetCrossRefGoogle Scholar
  93. 93.
    Brand, E., Gibson, J.F.: A doubly localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750 (2014)Google Scholar
  94. 94.
    Eckhardt, B.: Doubly localized states in plane Couette flow. J. Fluid Mech. 758, 1–4 (2014)CrossRefGoogle Scholar
  95. 95.
    Chantry, M., Willis, A.P., Kerswell, R.R.: Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow. Phys. Rev. Lett. 112, 164501 (2014)CrossRefGoogle Scholar
  96. 96.
    Mellibovsky, F., Eckhardt, B.: Takens-Bogdanov bifurcation of travelling-wave solutions in pipe flow. J. Fluid. Mech. 670, 96–129 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Beaume, C., Chini, G.P., Julien, K., Knobloch, E.: Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 043010 (2015)CrossRefGoogle Scholar
  98. 98.
    Beaume, C., Knobloch, E., Chini, G.P., Julien, K.: Modulated patterns in a reduced model of a transitional shear flow. Phys. Scr. 91, 024003 (2016)CrossRefGoogle Scholar
  99. 99.
    Wang, J., Gibson, J., Waleffe, F.: Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501 (2007)CrossRefGoogle Scholar
  100. 100.
    Blackburn, H.M., Hall, P., Sherwin, S.J.: Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726 (2013)Google Scholar
  101. 101.
    Deguchi, K., Hall, P.: The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99–112 (2014)MathSciNetCrossRefGoogle Scholar
  102. 102.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Comput. 7, 856–869 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems. SIAM J. Sci. Comput. 13, 631–644 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Lynch, R.E., Rice, J.R., Thomas, D.H.: Direct solution of partial difference equations by tensor product methods. Numerische Mathematik 6, 185–199 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    Dale, B., Haidvogel, D.B., Zang, T.: The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials. J. Comput. Phys. 30, 167–180 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Vitoshkin, H., Gelfgat, A.Yu.: On direct and semi-direct inverse of Stokes, Helmholtz and Laplacian operators in view of time-stepper-based Newton and Arnoldi solvers in incompressible CFD. Commun. Comput. Phys. 14, 1103–1119 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    Shi, L., Avila, M., Hof, B.: Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett. 110, 204502 (2013)CrossRefGoogle Scholar
  108. 108.
    Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D., Hof, B.: The onset of turbulence in pipe flow. Science 333, 192–196 (2011)CrossRefGoogle Scholar
  109. 109.
    Beaume, C.: Adaptive Stokes preconditioning for steady incompressible flows. Commun. Comput. Phys. 22, 494–516 (2017)MathSciNetCrossRefGoogle Scholar
  110. 110.
    Brynjell-Rahkola, M., Tuckerman, L.S., Schlatter, P., Henningson, D.S.: Computing optimal forcing using Laplace preconditioning. Commun. Comput. Phys. 22, 1508–1532 (2017)MathSciNetCrossRefGoogle Scholar
  111. 111.
    Fischer, P.: Nek5000. https://nek5000.mcs.anl.gov, Argonne National Laboratory, IL
  112. 112.
    Tuckerman, L.S.: Laplacian preconditioning for the inverse Arnoldi method. Commun. Comput. Phys. 18, 1336–1351 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Barkley, D., Tuckerman, L.S.: Stokes preconditioning for the inverse power method. In: Chattot, J.-J., Kutler, P., Flores, J. (eds.) 15th International Conference on Numerical Methods in Fluid Dynamics, pp. 75–76. Springer, Berlin (1997)Google Scholar
  114. 114.
    Tuckerman, L.S., Barkley, D.: Bifurcation analysis for timesteppers. In: Doedel, E., Tuckerman, L.S. (eds.) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, pp. 453–466. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  115. 115.
    Huepe, C., Tuckerman, L.S., Métens, S., Brachet, M.-E.: Stability and decay rates of nonisotropic attractive Bose–Einstein condensates. Phys. Rev. A 68, 023609 (2003)CrossRefGoogle Scholar
  116. 116.
    Gutknecht, M.H.: Variants of BiCGStab for matrices with complex spectrum. SIAM J. Sci. Comput. 14, 1020–1033 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Sleijpen, G.L.G., Fokkema, D.R.: BiCGstab for linear equations involving unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Anal. 1, 2000 (1993)MathSciNetzbMATHGoogle Scholar
  118. 118.
    Sonneveld, P., Van Gijzen, M.B.: IDR: a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations. SIAM J. Sci. Comput. 31, 1035–1062 (2008)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Laurette S. Tuckerman
    • 1
    Email author
  • Jacob Langham
    • 2
  • Ashley Willis
    • 3
  1. 1.Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI ParisPSL Research University, Sorbonne Université, Univ. Paris DiderotParisFrance
  2. 2.School of MathematicsUniversity of BristolBristolUK
  3. 3.School of Mathematics and StatisticsUniversity of SheffieldSheffieldUK

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